1. Kinetic and gravitational potential energy.
Let's consider a dropped ball again.
As I release it, its velocity is zero.
As it gets to the floor, it's attained a finite velocity.
From what we know already, we can figure out
the relationship between the height h from which
it was dropped and the speed v, which it had when
it hit the floor:
distance fallen h = average speed x time=(1/2)vxtime
v=time x g
Therefore time =v/g. Put it in the 1st equation:
h=(1/2)v(v/g) or gh=(1/2)v2

2. Kinetic and gravitational potential energy.
From F=ma and the fact that the gravitational
force is mg, we thus conclude that, for the dropped ball:
gh=(1/2)v2
or multiplying both sides by m
mgh=(1/2)mv2
at start just before it hits
We say that the ball had
Gravitational potential energy =mgh
At the start of its fall and that energy changed
form and became
Kinetic energy = (1/2)mv2
Just before the ball hit the floor

3. Conservation of Energy
In fact, a more general analysis shows that
at any moment during the fall, the sum
Gravitational energy + kinetic energy
stays the same. At the beginning of the fall,
the energy is all gravitational. At the bottom
of the fall it's all kinetic. In between, a gradual
change of form of the energy takes place, but the
total amount of energy stays the same.
This is what happens to the energy of water going over
falls or a dam. In hydropower systems, the energy
is stored in the water above the dam, and then converted
to kinetic energy as it goes over the dam. The kinetic
energy in the water is then used to do useful things
like generate electricity as we will discuss later.

4. Hoover Dam on
the Colorado river
In Nevada
Height of the
water at the
top above the
bottom= 180m
Volume of water in
The lake above the
Dam= 32 km3

5. Review
velocity is (change in position)/time elapsed
acceleration is (change in velocity/time elapsed
total force =mass x acceleration
gravitational force = Mass x g down
From F=ma for falling object
(1/2)mv2 + mgh = constant
kinetic gravitational
energy potential energy
Conservation of energy when the only forces
acting to accelerate the body
are gravitational. When other forces are
acting there are more terms on the left hand
side.

6. If I dropped a ball from a height of 2 meters,
what was its speed just before it hit the floor?
A. 9.8x2 m/s
B. 9.8/2 m/s
_____
C. √9.8x2 m/s
D. √9.8x4 m/s
E. 9.8x4 m/s

7. Units of force:
Since total force is mass x acceleration
a suitable metric unit is
1kg meter/second2 called 1 Newton
In the British system, the force unit chosen is
the pound. The British mass unit (not used
much) is a slug and
1 pound = 1slug ft/second2
1 Newton = .225 pounds
We will tell you this conversion value if you
need it. (Not in Appendix B).

8. Weight:
The magnitude of the gravitational force on an
object is called its weight.
Therefore the weight of an object of mass M
is always Mg near the surface of the earth.
Because g varies with altitude, the weight can
be different in different places for a given object
but the mass does not change.
The units of weight are therefore force units.

9. Units of energy:
We can figure out the units of energy
from the expression for the gravitational
potential energy:
mgh= (a force) times (a length)‏
So metric units of force are Newtonsxmeters
which we call joules.
1 joule = 1 newton meter
British units of energy can then be
1 ft-pound
Confusingly, this is NOT a British Thermal Unit (btu)‏
1Btu= 1055Joules=778 ft-lb

10. Again, you do not have to remember the conversion
factors which are listed in Chapter 3, Appendix
B and inside the back cover of your book.
However you are expected to be able to use the
conversion factors to change a quantity from
one kind of units to another.
It is also important to always use the right
KIND of units: energy units for energy, force units
for force, etc.

11. Energy transfers between objects.
Consider holding up the book again.
Suppose its mass is 1kg and that I
slowly raise it 1 meter. How much
gravitational potential energy did it gain?
A. 1 joule
B. 9.8 joules
C. 1 newton
D. 9.8 newtons

12. OK it gained 9.8 joules. Since energy is
conserved, where did this energy
come from?
A. The kinetic energy of the book.
B. chemical energy in my body
C. the action of the friction of the air.
D. chemical energy in the book.

13. Now I want to concentrate on how the
energy was transferred from my body
to the book.
I pressed my hand against the bottom
of the book, and produced an upward
force which was just slightly larger than
the downward gravitational
force mg so that the book slowly
moved up a distance h. In magnitude the
resulting gain in energy was
mg x h which is equal to
(the upward force with which I pushed
with my hand)‏
times
(the distance through which I pushed the
book along the direction of the force. )‏

14. This is an example of the transfer of
energy from one object to another
through the performance of WORK.
When one object exerts a force F on
another and the second object
moves along the direction of
the force a distance d, then we say that
the first object has done work Fd on
the second object, resulting in energy
transfer Fd from the first to the second object.

15. Notes about work:
Work has the units of energy but it is always
an amount of energy transferred from
one object to another and NOT the amount
of energy in a object. After the energy
is transferred through the performance of
work, the energy ends up in the second
object in one of the forms we will be discussing
such as gravitational potential, kinetic,
thermal, chemical or electrical potential energy.
In the formula work=Fd, d is the distance
ALONG THE FORCE. This may not be the same
as the distance which the contact point between
one object and the other moves.

16. Suppose you slowly carry an object weighing
10 pounds up a flight of stairs consisting of
20 steps. Each step is 1/2 feet high and 1 foot
wide. How much work did you perform on the
object?
A ft-pounds
B. 200 ft-pounds
C. (5)1/2x100 ft-pounds
D. 0

17. You carry the same 10 pound object
100 ft down the horizontal hall.
During the time between the time
after you picked it up and before you
put it down, how much work did you
perform on the object? (Suppose
you walked slowly at constant
velocity).
A ft-pounds
B. 0
C. 500 ft-pounds
D. insufficient information is given

18. From a zero velocity start, I push a cart
along a frictionless track with a constant
force for 1/2 meter and then let it go.
Its velocity after I release it is observed
to be 1m/s and the mass of the cart is
1/4 kg. What was the magnitude of the
force with which I pushed it?
A. 0.25N
B. 2.45N
C. 0.5N
D. 0
Question 6

19. I throw a 1/2 kg object straight up.
During the throw, I exert a constant
force upward on the object while it
moves upward through 1/2 meter.
Just after I release the object, it is observed
to have an upward velocity of magnitude
(speed) 2 meters per second. What was the
magnitude of the force which I exerted on the
object during the throw?
A. 4.8 N
B. 6.9 N
C. 0.25N
D. 0.5N
Question 7

20. Answer: B
The point here is that the object ends up with
two forms of additional energy, kinetic and
gravitational potential:
Fd = (1/2)mv2 +mgd
Work gain in gain in gravitational
Done kinetic potential energy
energy
F(1/2 m)=(1/2)(1/2kg)(2m/s)2 +(1/2kg)(9.8m/s2)(1/2 m)
Divide both sides by ½ meter and simplify
F=2kgm/s kgm/s2 = 6.9 newtons

21. OK that was the force exerted by
my hand on the ball. What was the
total force on the ball during the toss?
(1/2 kg ball, pushed up 1/2m, final
speed 2 m/s).
A newtons
B. 4.9 newtons
C. 2 newtons
D. 0
Question 11

22. The total force on the ball was 2N upward and
the velocity of the ball was 2m/s as it left my hand.
The mass of the ball was 1/4 kg.
How long (how many seconds) was a pushing
the ball upward before it left my hand?
A seconds
B. 0.5 seconds
C seconds
D seconds
E seconds

23. Summary:
Energy is transferred from one object to another
through the performance of work. The amount
of work done by one object on another is
Work done = F x d
where F is the force which the first object exerts
on the second and d is the distance, along the
direction of the force, through which the force
acts. The work done is the amount of energy
which is transferred from the first object to
the second. The energy ends up in one of the forms
we are discussing: kinetic, gravitational potential,
thermal. It is not meaningful to talk of work IN a
body. If the force acting is opposite in direction
to the direction through which it acts, the work
done is negative and energy is extracted from
the second object.